The present application relates to the stability of simultaneous (or parallel) machining, where multiple tools operate at different spindle speeds on the same work-piece. For instance and as used herein, the term “simultaneous machining” (SM) encompasses multiple conventional milling spindles, single milling cutters with non-uniformly distributed cutter flutes, and other machining applications that are characterized, at least in part, by non-uniform pitch cutters, including single spindle machining applications that are characterized by such non-uniformity. Simultaneous machining operations (as defined above), as opposed to conventional single tool machining (STM) (also known as serial process), are generally more time efficient. It is also known that the dynamics of SM applications are considerably more complex. When metal removal rates are maximized, the dynamic coupling among the cutting tools, the work-piece and the machine tool(s) become very critical and regenerative forces are generally more pronounced. The dynamic stability repercussions of such settings are poorly understood at present, even in the mathematics community. In fact, there is no analytical mechanism available to assess such regenerative forces in SM applications and no evidence in the literature addressing the stability of simultaneous machining.
It is well known from numerous investigations that conventional single tool machining (STM) introduces significant stability issues, e.g., when STM operations are optimized. Stability issues are even more pronounced in more complex SM applications and systems. In the absence of a solid mathematical methodology to study and/or address SM chatter, the existing industrial practice is sub-optimal and is generally guided by trial-and-error or ad hoc procedures. Accordingly, there is room for much-needed improvement in the machining field.
Optimum machining aims to maximize the material removal rate, while maintaining a sufficient stability margin to assure the surface quality. Machine tool instability associated with machining applications primarily relates to ‘chatter’. As accepted in the manufacturing community, there are two groups or types of machine tool chatter: regenerative and non-regenerative. See, e.g., J. Tlusty, Machine Dynamics, Handbook of High Speed Machining Technology: Chapman and Hall, NY, 1985. Regenerative chatter occurs due to the periodic tool passing over the undulations on the previously cut surface, and non-regenerative chatter is associated with mode coupling among the existing modal oscillations. The methods and systems of the present disclosure are primarily aimed at mapping and/or control of regenerative chatter and, for purposes of the detailed description which follows, the term “chatter” refers to “regenerative chatter” unless otherwise noted.
Generally, in order to prevent the onset of chatter, a manufacturer/system operator must select appropriate operational parameters, e.g., chip loads and spindle speeds. Existing studies on machine tool stability address conventional single-tool machining processes. They are inapplicable, however, to SM because of the substantial differences in the underlying mathematics. Indeed, simultaneous machining gives rise to a complex mathematical characterization known as “parametric quasi-polynomials with multiple delay terms”. There is no practical methodology known at this point, to resolve the complete stability mapping for such constructs.
Machine tool chatter is an undesired engineering phenomenon. Its negative effects on the surface quality, tool life and other operational parameters/results are well known. Starting with early reported works[7-10], many researchers meticulously addressed the issues of modeling, dynamic progression, structural reasoning and stability limit aspects of this seemingly straightforward and very common behavior. See, M. E. Merchant, “Basic Mechanics of the Metal-Cutting Process,” ASME Journal of Applied Mechanics, vol. 66, pp. A-168, 1944; S. Doi and S. Kato, “Chatter Vibration of Lathe Tools,” Transactions of ASME, vol. 78, pp. 1127, 1956; S. A. Tobias, Machine Tool Vibration: Wiley, NY, 1961; and J. Tlusty, L. Spacek, M. Polacek, and O. Danek, Selbsterregte Schwingungen an Werkzeugmaschinen: VEB Verlag Technik, Berlin, 1962. Further research focused on the particulars of parameter selections in machining to avoid the build-up of these undesired oscillations and on the analytical predictions of chatter. See, J. Tlusty, Machine Dynamics, Handbook of High Speed Machining Technology: Chapman and Hall, NY, 1985; J. Tlusty, L. Spacek, M. Polacek, and O. Danek, Selbsterregte Schwingungen an Werkzeugmnaschinen: VEB Verlag Technik, Berlin, 1962; H. E. Merritt, “Theory of Self Excited Machine Tool Chatter,” Journal of Engineering for Industry, pp. 447, 1965; and R. L. Kegg, “Cutting Dynamics in Machine Tool Chatter,” Journal of Engineering for Industry, pp. 464, 1965. Most commonly, chatter research has focused on the conventional single tool machining (STM).
Generally, the principle aim of machining applications is to increase the metal removal rate while avoiding the onset of chatter. See, e.g., Y. Altintas and E. Budak, “Analytical Prediction of Stability Lobes in Milling,” Annals of the CIRP, vol. 44, 1995; S. Smith and J. Tlusty, “Efficient Simulation Program for Chatter in Milling,” Annals of the CIRP, vol. 42, 1993; and S. Smith and J. Tlusty, “Update on High Speed Milling Dynamics,” Tran. ASME, J. Of Engineering for Industry, 1990. A natural progressive trend is to increase the productivity through simultaneous (or parallel) machining. Ideally, this process can be further optimized by determining the best combination of chip loads and spindle speeds with the constraint of chatter instability. For SM, however, multiple spindle speeds, which cross-influence each other, create governing differential equations with multiple time delay terms. Their characteristic equations are known in mathematics as “quasi polynomials with multiple time delays”. Multiplicity of the delays present enormously more complicated problems compared with the conventional single-tool machining (STM) chatter and have heretofore prevented the mapping and/or control of SM applications for optimization purposes.
For background purposes, the basics of STM chatter dynamics are reviewed herein. In this regard, reference is specifically made to the text “Machine Dynamics, Handbook of High Speed Machining Technology,” J. Tlusty, Chapman and Hall, NY, 1985. For illustrative purposes, reference is made to FIG. 1 which relates to an exemplary orthogonal turning process. The underlying mechanism for regenerative chatter is quite simple to state. The desired (and nominal) chip thickness, ho, is considered constant. The tool actually cuts the chip from the surface, which is created during the previous pass. The process-generated cutting force, F, is realistically assumed to be proportional to the dynamic chip thickness, h(t). Such force carries the signature of y(t)−y(t−τ), where y(t) is the fluctuating part of the chip thickness at time t (so called “offset chip thickness” from the nominal value h0), and τ (sec) is the period of successive passages of the tool, which is equal to 60/N, where N is the spindle speed (RPM).
The block diagram in FIG. 2 gives a classical causality representation of the dynamics for this orthogonal turning operation. Nominal chip thickness, ho, is disturbed by the undulating offset chip thickness, y. These undulations create driving forces for the y dynamics τ sec later during the next passage and thus the attribute “regenerative” chatter. G(s) is the transfer function between the cutting force, F, and y. For the sake of streamlining the analysis, a single-degree-of-freedom cutting dynamics is taken into account instead of higher-degree-of-freedom and more complex models. The work-piece and its rotational axis are considered to be rigidly fixed and the only tool flexibility is taken in the radial direction. In FIG. 2 the following causal relations are incorporated
Cutting forceF(t) = Cbh(t)(1)Actual chip thicknessh(t) = ho − y(t) + y(t − τ)(2)where b is the chip width which is user selected and assumed constant, C is the cutting force constant, τ [sec] is the period of one spindle revolution, τ=60/N, N [RPM].
Assuming that the force-displacement transfer function G(s) is linear, the entire cutting mechanism described by FIG. 1 is linear. Cutting would be at equilibrium if y=0, which means an ideal cut with no waviness. The cutting force, F, remains constant and the tool support structure (i.e., k, c) yields a static deflection throughout the cutting. This equilibrium is called “stable” or “asymptotically stable” if the loop characteristic equation of the block in FIG. 2 [Eq. 3] has all its roots on the stable left half plane.1+(1−e−τs)bCG(s)=0  (3)This equation is transcendental and it possesses infinitely many finite characteristic roots, all of which have to be taken into account for stability. Although the problem looks prohibitively complex, the complete stability map is obtainable for single delay cases (i.e., STM). It is clear that the selection of b and τ influences the stability of the system considerably. The complete stability map of this system in b and τ domain are the well known “chatter stability lobes” as shown in FIG. 3 (using the parameters from N. Olgac and M. Hosek, “A New Perspective and Analysis for Regenerative Machine Tool Chatter,” International Journal of Machine Tools & Manufacture, vol. 38, pp. 783-798, 1998). There are several such lobes marked on FIG. 3 which represent the (b, τ) plane mapping of the dynamics with dominant characteristic root at Re(s)=−a. When a=0 (the thick line), these curves show the well known stability lobes which form the stability constraints for any process optimization. For instance, if the removal rate is increased (e.g., by increasing the chip width, b), chatter instability is ultimately encountered which is obviously undesirable and unacceptable.
There are generally two cutting conditions under the control of the machinist: τ, which is the inverse of the spindle speed (60/N), and b, the chip width. The other parameters, i.e., C, G(s), represent the existing cutting characteristics, which are considered to remain unchanged. The open loop transfer function G(s) typically manifests high-impedance, damped, and stable dynamic behavior.
Certain selections of b and τ(=60/N) can introduce marginal stability to the system, as shown in FIG. 3. At these operating points, the characteristic equation (3) possesses a pair of imaginary roots, i as dominant roots. As such, the complete system is resonant at. That is, the entire structure mimics a spring-mass resonator (i.e., a conservative system) at the respective frequency of marginal stability (also known as “chatter frequency”). The desirable operating point should lie in the shaded region, marked as “stable” in FIG. 3.
For appropriate operational margins, it is desirable to select the cutting parameters (b, N) sufficiently away from the chatter stability bounds. Conventional terminology alluding to this feature is called the “stability margin”, which refers to the a=−Re (dominant characteristic root). The bigger the value of a, the higher the “chatter rejection speed”, therefore, the better the surface quality. A set of operating points are shown in FIG. 3, where a=5, 10 and 15, illustrating the distribution of the loci where the chatter rejection speed is constant. Optimum working conditions are reached by increasing b and N up to the physical limitations, provided that a desirable stability margin (i.e., a) is guaranteed. FIG. 3 also represents a unique declaration in the machine tool literature as “equal stability margin” lobes.
Regenerative Chatter in Simultaneous Machining
The functional block diagram in FIG. 2 expands in dimension by the number of tools involved in the case of simultaneous machining, as depicted in FIGS. 4a and 4b for two exemplary operations. The flexures are shown at a single point and in a 1-D sense to symbolize the most complex form of the restoring forces in 3-D. Spindle speeds and their directions are also selected symbolically for purposes of describing/reflecting the types of operations at issue.
A crucial difference between STM and SM operations is the coupling among the individual tool-workpiece interfaces (e.g., coupling among multiple milling spindles and/or coupling among non-uniformly distributed cutter flutes on a single milling cutter), either through the flexible work-piece (as in FIG. 4a) or the machine tool compliance characteristics (as in FIG. 4b), or both. Clearly, the chip load at tool i (i=1 . . . n) which carries the signature of yi(t)−yi(t−τi), will influence the dynamics at tool j. This is a cross-regenerative effect, which implies that the state of the ith tool one revolution earlier (i.e., τi seconds) affects the present dynamics of the jth tool. Consequently, the dynamics of tool j will reflect the combined regenerative effects from all the tools, including itself. Assuming these tool-to-tool interfaces are governed under linear relations, the overall dynamics become a truly cross-coupled linear multiple time-delay system. The cutting force directions are time varying (and periodic) in nature. However, as contemplated in the literature, the fundamental elements of these forces may be addressed in their Fourier expansions. See, e.g., Y. Altintas, S. Engin, and E. Budak, “Analytical Stability Prediction and Design of Variable Pitch Cutters,” ASME Journal of Manufacturing Science and Engineering, vol. 121, pp. 173-178, 1999. As such, the linear-time-invariant nature is maintained in the overall dynamics.
The stability of multiple delay systems is poorly known in the mathematics community. There is no simple extension of the conventional (STM) stability treatment to multiple spindle and/or non-uniformly distributed flutes on a single cutter, and consequently to multiple delay cases. Accordingly, the ability to predict and/or control chatter stability in such SM systems/applications is essentially non-existent.
It is noted that prior art investigations which perform time domain simulations, such as described by Y. Altintas, S. Engin and E. Budak in “Analytical Stability Prediction and Design of Variable Pitch Cutters,” ASME Journal of Manufacturing Science and Engineering, Vol. 121, pp. 173-78 (1999), are non-analogous operations as compared to the systems and methods of the present disclosure. First, the foregoing investigations are performed point-by-point in (τ1, τ2) space and, therefore, the application is computationally overwhelming. Second, the foregoing investigations are of a numerical (as opposed to analytical) nature. Other reported mathematical investigations also declare strict limitations and restrictions to their pursuits. For example, in C. S. Hsu, “Application of the Tau-Decomposition Method to Dynamical Systems Subjected to Retarded Follower Forces,” ASME Journal of Applied Mechanics, vol. 37, pp. 258-266, 1970, it is claimed that it is very difficult to utilize Pontryagin's Theorem for the stability analysis when n>1 and det(G(s)) invites higher degree terms of ‘s’ than one. In a further publication, the same problem is handled using simultaneous nonlinear equation solvers, but such approach is only able to treat systems with Eq (4) (see below) in a+b e−τ1s+c e−τ2s=0 construct with a, b, c being constants. See, J. K. Hale and W. Huang, “Global Geometry of the Stable Regions for Two Delay Differential Equations,” Journal of Mathematical Analysis and Applications, vol. 178, pp. 344-362, 1993. This approach imposes an obvious and very serious restriction for the problem at hand. Further prior art teachings tackle a limited sub-class of equation (4) with detG(s) of degree 2, but without a damping term, and the disclosed approach is not expandable to damped cases. See, S.-I. Niculescu, “On Delay Robustness Analysis of a Simple Control Algorithm in High-Speed Networks,” Automatica, vol. 38, pp. 885-889, 2002; and G. Stepan, Retarded Dynamical Systems: Stability and Characteristic Function. New York: Longman Scientific & Technical, co-publisher John Wiley & Sons Inc., US., 1989. The foregoing references also consider the parameter vector (B) as fixed.
Thus, despite efforts to date, a need exists for systems and methods that facilitate chatter stability mapping and/or control in simultaneous machining applications. A further need exists for systems and methods that permit optimized and/or enhanced STM operations at least in part based on chatter stability mapping and/or control. These and other needs and objectives are met by the disclosed systems and methods described herein.